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arXiv:2012.02063 · Bulletin of the London Mathematical Society

A geometric approach to Wigner‐type theorems

Mark Pankov, Thomas Vetterlein

Published May 16, 2021• 5 citations
arXivHTMLPDF

Abstract

Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

DOI
10.1112/blms.12517

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Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.
Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let H1\mathcal{H}_1H1​ and H2\mathcal{H}_2H2​ be Hilbert spaces, and let kkk be a positive integer such that 2≤k<12dim⁡(H1)2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)2≤k<21​dim(H1​). We show that if an injective map ψ:Pk(H1)→Pk(H2)\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)ψ:Pk​(H1​)→Pk​(H2​) satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let M\mathcal{M}M be a type II1_11​ or type II∞_\infty∞​ factor, let τM\tau_\mathcal{M}τM​ be a faithful normal semi-finite tracial weight of M\mathcal MM, and let $0
Abstract Let HHH be a Hilbert space and P(H)P(H)P(H) be the projective space of all quantum pure states. Wigner’s theorem states that every bijection ϕ ⁣:P(H)→P(H)\phi \colon P(H)\to P(H)ϕ:P(H)→P(H) that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator U ⁣:H→HU\colon H\to HU:H→H. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle π2\frac{\pi }{2}2π​ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha $ in both directions, provided that 0<α≤π40 < \alpha \leq \frac{\pi }{4}0<α≤4π​ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy π4<α<π2\frac{\pi }{4} < \alpha < \frac{\pi }{2}4π​<α<2π​, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s.
It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two nnn-dimensional Hilbert spaces, n≥3n\ge 3n≥3, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) nnn-tuples of lines in an nnn-dimensional Hilbert space VVV by F⊥(V)\mathbb{F}^{\perp}(V)F⊥(V) (respectively F(V)\mathbb{F}(V)F(V)) and, for partitions πππ of the set {1..n}\{1..n\}{1..n}, call two tuples πππ-linked if the spans along πππ-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps F⊥(Cn)→F(Cn)\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)F⊥(Cn)→F(Cn), n≥3n\ge 3n≥3 respecting πππ-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the F(Cn)\mathbb{F}(\mathbb{C}^n)F(Cn)-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple (ℓi)i(\ell_i)_i(ℓi​)i​ of lines into ((⨁j≠iℓj)⊥)i\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i((⨁j=i​ℓj​)⊥)i​.